FastMath dist4

Percentage Accurate: 87.5% → 98.5%

Time: 3.1s

Alternatives: 11

Speedup: 1.2×

• 33.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
}

Python

def code(d1, d2, d3, d4):
	return (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = (((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(d4, d1, \left(d2 - d3\right) \cdot d1\right) - d1 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d2 + \left(d4 - d1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)) INFINITY)
   (- (fma d4 d1 (* (- d2 d3) d1)) (* d1 d1))
   (* d1 (+ d2 (- d4 d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (((((d1 * d2) - (d1 * d3)) + (d4 * d1)) - (d1 * d1)) <= ((double) INFINITY)) {
		tmp = fma(d4, d1, ((d2 - d3) * d1)) - (d1 * d1);
	} else {
		tmp = d1 * (d2 + (d4 - d1));
	}
	return tmp;
}

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(d1 * d2) - Float64(d1 * d3)) + Float64(d4 * d1)) - Float64(d1 * d1)) <= Inf)
		tmp = Float64(fma(d4, d1, Float64(Float64(d2 - d3) * d1)) - Float64(d1 * d1));
	else
		tmp = Float64(d1 * Float64(d2 + Float64(d4 - d1)));
	end
	return tmp
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[N[(N[(N[(N[(d1 * d2), $MachinePrecision] - N[(d1 * d3), $MachinePrecision]), $MachinePrecision] + N[(d4 * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(d4 * d1 + N[(N[(d2 - d3), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision] - N[(d1 * d1), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d2 + N[(d4 - d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1)) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      5. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. *-commutative N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(d1 \cdot d4 + \left(d1 \cdot d2 - d1 \cdot d3\right)\right)} - d1 \cdot d1 \]

      8. *-commutative N/A

         \[\leadsto \left(\color{blue}{d4 \cdot d1} + \left(d1 \cdot d2 - d1 \cdot d3\right)\right) - d1 \cdot d1 \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(d4, d1, d1 \cdot d2 - d1 \cdot d3\right)} - d1 \cdot d1 \]

      10. distribute-lft-out-- N/A

          \[\leadsto \mathsf{fma}\left(d4, d1, \color{blue}{d1 \cdot \left(d2 - d3\right)}\right) - d1 \cdot d1 \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(d4, d1, \color{blue}{\left(d2 - d3\right) \cdot d1}\right) - d1 \cdot d1 \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(d4, d1, \color{blue}{\left(d2 - d3\right) \cdot d1}\right) - d1 \cdot d1 \]

      13. lower--.f64 100.0

          \[\leadsto \mathsf{fma}\left(d4, d1, \color{blue}{\left(d2 - d3\right)} \cdot d1\right) - d1 \cdot d1 \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(d4, d1, \left(d2 - d3\right) \cdot d1\right)} - d1 \cdot d1 \]

   if +inf.0 < (-.f64 (+.f64 (-.f64 (*.f64 d1 d2) (*.f64 d1 d3)) (*.f64 d4 d1)) (*.f64 d1 d1))

   1. Initial program 0.0%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      11. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      12. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

      14. pow2 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

      15. associate--l+ N/A

          \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d1 \cdot d3\right)\right)} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{d3 \cdot d1}\right)\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d3\right)\right) \cdot d1} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot d3\right)} \cdot d1 + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   3. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right)} \]

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot \left(d4 - d1\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 - d1\right)}\right) \]

      4. lift--.f64 88.1

         \[\leadsto d1 \cdot \left(d2 + \left(d4 - \color{blue}{d1}\right)\right) \]

   6. Applied rewrites 88.1%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (fma (- d3) d1 (fma d2 d1 (* d1 (- d4 d1)))))

C

double code(double d1, double d2, double d3, double d4) {
	return fma(-d3, d1, fma(d2, d1, (d1 * (d4 - d1))));
}

Julia

function code(d1, d2, d3, d4)
	return fma(Float64(-d3), d1, fma(d2, d1, Float64(d1 * Float64(d4 - d1))))
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[((-d3) * d1 + N[(d2 * d1 + N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   7. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

   8. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

   9. mul-1-neg N/A

      \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   10. associate-*r* N/A

       \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   11. +-commutative N/A

       \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

   12. *-commutative N/A

       \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

   13. associate-+r+ N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

   14. pow2 N/A

       \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

   15. associate--l+ N/A

       \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right)} \]

   16. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d1 \cdot d3\right)\right)} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   17. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(\color{blue}{d3 \cdot d1}\right)\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   18. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d3\right)\right) \cdot d1} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   19. mul-1-neg N/A

       \[\leadsto \color{blue}{\left(-1 \cdot d3\right)} \cdot d1 + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

3. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right)} \]
4. Add Preprocessing

Alternative 3: 85.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 1.7 \cdot 10^{+63}:\\ \;\;\;\;d1 \cdot \left(\left(d2 - d3\right) - d1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(d4 + d2\right) - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 1.7e+63) (* d1 (- (- d2 d3) d1)) (* (- (+ d4 d2) d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.7e+63) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = ((d4 + d2) - d3) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 1.7d+63) then
        tmp = d1 * ((d2 - d3) - d1)
    else
        tmp = ((d4 + d2) - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 1.7e+63) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = ((d4 + d2) - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 1.7e+63:
		tmp = d1 * ((d2 - d3) - d1)
	else:
		tmp = ((d4 + d2) - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 1.7e+63)
		tmp = Float64(d1 * Float64(Float64(d2 - d3) - d1));
	else
		tmp = Float64(Float64(Float64(d4 + d2) - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 1.7e+63)
		tmp = d1 * ((d2 - d3) - d1);
	else
		tmp = ((d4 + d2) - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 1.7e+63], N[(d1 * N[(N[(d2 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d4 + d2), $MachinePrecision] - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 1.6999999999999999e63

   1. Initial program 88.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 84.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   if 1.6999999999999999e63 < d4

   1. Initial program 81.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 90.2

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   4. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d4 \leq 5.5 \cdot 10^{+65}:\\ \;\;\;\;d1 \cdot \left(\left(d2 - d3\right) - d1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d4 5.5e+65) (* d1 (- (- d2 d3) d1)) (* (- d4 d3) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 5.5e+65) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d4 <= 5.5d+65) then
        tmp = d1 * ((d2 - d3) - d1)
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d4 <= 5.5e+65) {
		tmp = d1 * ((d2 - d3) - d1);
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d4 <= 5.5e+65:
		tmp = d1 * ((d2 - d3) - d1)
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d4 <= 5.5e+65)
		tmp = Float64(d1 * Float64(Float64(d2 - d3) - d1));
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d4 <= 5.5e+65)
		tmp = d1 * ((d2 - d3) - d1);
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d4, 5.5e+65], N[(d1 * N[(N[(d2 - d3), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision], N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d4 < 5.4999999999999996e65

   1. Initial program 88.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 84.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 84.5%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   if 5.4999999999999996e65 < d4

   1. Initial program 81.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 90.4

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   5. Taylor expanded in d2 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d2 - d1\right)\\ \mathbf{if}\;d4 \leq 1.3 \cdot 10^{-235}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d4 \leq 2.5 \cdot 10^{-91}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{elif}\;d4 \leq 2.4 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(d4 - d3\right) \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d2 d1))))
   (if (<= d4 1.3e-235)
     t_0
     (if (<= d4 2.5e-91)
       (* d1 (- d2 d3))
       (if (<= d4 2.4e+65) t_0 (* (- d4 d3) d1))))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d1);
	double tmp;
	if (d4 <= 1.3e-235) {
		tmp = t_0;
	} else if (d4 <= 2.5e-91) {
		tmp = d1 * (d2 - d3);
	} else if (d4 <= 2.4e+65) {
		tmp = t_0;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: tmp
    t_0 = d1 * (d2 - d1)
    if (d4 <= 1.3d-235) then
        tmp = t_0
    else if (d4 <= 2.5d-91) then
        tmp = d1 * (d2 - d3)
    else if (d4 <= 2.4d+65) then
        tmp = t_0
    else
        tmp = (d4 - d3) * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d2 - d1);
	double tmp;
	if (d4 <= 1.3e-235) {
		tmp = t_0;
	} else if (d4 <= 2.5e-91) {
		tmp = d1 * (d2 - d3);
	} else if (d4 <= 2.4e+65) {
		tmp = t_0;
	} else {
		tmp = (d4 - d3) * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = d1 * (d2 - d1)
	tmp = 0
	if d4 <= 1.3e-235:
		tmp = t_0
	elif d4 <= 2.5e-91:
		tmp = d1 * (d2 - d3)
	elif d4 <= 2.4e+65:
		tmp = t_0
	else:
		tmp = (d4 - d3) * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d2 - d1))
	tmp = 0.0
	if (d4 <= 1.3e-235)
		tmp = t_0;
	elseif (d4 <= 2.5e-91)
		tmp = Float64(d1 * Float64(d2 - d3));
	elseif (d4 <= 2.4e+65)
		tmp = t_0;
	else
		tmp = Float64(Float64(d4 - d3) * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d2 - d1);
	tmp = 0.0;
	if (d4 <= 1.3e-235)
		tmp = t_0;
	elseif (d4 <= 2.5e-91)
		tmp = d1 * (d2 - d3);
	elseif (d4 <= 2.4e+65)
		tmp = t_0;
	else
		tmp = (d4 - d3) * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d4, 1.3e-235], t$95$0, If[LessEqual[d4, 2.5e-91], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d4, 2.4e+65], t$95$0, N[(N[(d4 - d3), $MachinePrecision] * d1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d4 < 1.3e-235 or 2.49999999999999999e-91 < d4 < 2.4000000000000002e65

   1. Initial program 88.5%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 82.0

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 82.0%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   5. Taylor expanded in d2 around inf

      \[\leadsto d1 \cdot \left(d2 - d1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 57.7%

      \[\leadsto d1 \cdot \left(d2 - d1\right) \]

   if 1.3e-235 < d4 < 2.49999999999999999e-91

   1. Initial program 91.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 99.4

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 72.5

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 72.5%

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]

   if 2.4000000000000002e65 < d4

   1. Initial program 88.5%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around 0

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 + d4\right) - d3\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot \color{blue}{d1} \]

      3. lower--.f64 N/A

         \[\leadsto \left(\left(d2 + d4\right) - d3\right) \cdot d1 \]

      4. +-commutative N/A

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

      5. lower-+.f64 78.4

         \[\leadsto \left(\left(d4 + d2\right) - d3\right) \cdot d1 \]

   4. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\left(\left(d4 + d2\right) - d3\right) \cdot d1} \]

   5. Taylor expanded in d2 around 0

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
   6. Step-by-step derivation

   7. Applied rewrites 52.5%

      \[\leadsto \left(d4 - d3\right) \cdot d1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d4 - d1\right)\\ t_1 := d1 \cdot \left(d2 - d3\right)\\ \mathbf{if}\;d4 \leq 1.3 \cdot 10^{-235}:\\ \;\;\;\;d1 \cdot \left(d2 - d1\right)\\ \mathbf{elif}\;d4 \leq 9.6 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d4 \leq 5.8 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d4 \leq 1.18 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (let* ((t_0 (* d1 (- d4 d1))) (t_1 (* d1 (- d2 d3))))
   (if (<= d4 1.3e-235)
     (* d1 (- d2 d1))
     (if (<= d4 9.6e-35)
       t_1
       (if (<= d4 5.8e+32) t_0 (if (<= d4 1.18e+95) t_1 t_0))))))

C

double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d4 - d1);
	double t_1 = d1 * (d2 - d3);
	double tmp;
	if (d4 <= 1.3e-235) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 9.6e-35) {
		tmp = t_1;
	} else if (d4 <= 5.8e+32) {
		tmp = t_0;
	} else if (d4 <= 1.18e+95) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = d1 * (d4 - d1)
    t_1 = d1 * (d2 - d3)
    if (d4 <= 1.3d-235) then
        tmp = d1 * (d2 - d1)
    else if (d4 <= 9.6d-35) then
        tmp = t_1
    else if (d4 <= 5.8d+32) then
        tmp = t_0
    else if (d4 <= 1.18d+95) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double t_0 = d1 * (d4 - d1);
	double t_1 = d1 * (d2 - d3);
	double tmp;
	if (d4 <= 1.3e-235) {
		tmp = d1 * (d2 - d1);
	} else if (d4 <= 9.6e-35) {
		tmp = t_1;
	} else if (d4 <= 5.8e+32) {
		tmp = t_0;
	} else if (d4 <= 1.18e+95) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	t_0 = d1 * (d4 - d1)
	t_1 = d1 * (d2 - d3)
	tmp = 0
	if d4 <= 1.3e-235:
		tmp = d1 * (d2 - d1)
	elif d4 <= 9.6e-35:
		tmp = t_1
	elif d4 <= 5.8e+32:
		tmp = t_0
	elif d4 <= 1.18e+95:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(d1, d2, d3, d4)
	t_0 = Float64(d1 * Float64(d4 - d1))
	t_1 = Float64(d1 * Float64(d2 - d3))
	tmp = 0.0
	if (d4 <= 1.3e-235)
		tmp = Float64(d1 * Float64(d2 - d1));
	elseif (d4 <= 9.6e-35)
		tmp = t_1;
	elseif (d4 <= 5.8e+32)
		tmp = t_0;
	elseif (d4 <= 1.18e+95)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	t_0 = d1 * (d4 - d1);
	t_1 = d1 * (d2 - d3);
	tmp = 0.0;
	if (d4 <= 1.3e-235)
		tmp = d1 * (d2 - d1);
	elseif (d4 <= 9.6e-35)
		tmp = t_1;
	elseif (d4 <= 5.8e+32)
		tmp = t_0;
	elseif (d4 <= 1.18e+95)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := Block[{t$95$0 = N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d4, 1.3e-235], N[(d1 * N[(d2 - d1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d4, 9.6e-35], t$95$1, If[LessEqual[d4, 5.8e+32], t$95$0, If[LessEqual[d4, 1.18e+95], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d4 < 1.3e-235

   1. Initial program 88.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 80.1

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 80.1%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   5. Taylor expanded in d2 around inf

      \[\leadsto d1 \cdot \left(d2 - d1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 56.8%

      \[\leadsto d1 \cdot \left(d2 - d1\right) \]

   if 1.3e-235 < d4 < 9.6000000000000005e-35 or 5.80000000000000006e32 < d4 < 1.17999999999999998e95

   1. Initial program 90.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 93.6

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 93.6%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 68.4

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]

   if 9.6000000000000005e-35 < d4 < 5.80000000000000006e32 or 1.17999999999999998e95 < d4

   1. Initial program 82.6%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      11. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      12. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

      14. pow2 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

      15. associate--l+ N/A

          \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d1 \cdot d3\right)\right)} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{d3 \cdot d1}\right)\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d3\right)\right) \cdot d1} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot d3\right)} \cdot d1 + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   3. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right)} \]

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot \left(d4 - d1\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 - d1\right)}\right) \]

      4. lift--.f64 83.3

         \[\leadsto d1 \cdot \left(d2 + \left(d4 - \color{blue}{d1}\right)\right) \]

   6. Applied rewrites 83.3%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d1\right)\right)} \]

   7. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   8. Step-by-step derivation

      1. lift--.f64 67.7

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   9. Applied rewrites 67.7%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -9 \cdot 10^{-39}:\\ \;\;\;\;d1 \cdot \left(d2 - d3\right)\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -9e-39) (* d1 (- d2 d3)) (* d1 (- d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9e-39) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-9d-39)) then
        tmp = d1 * (d2 - d3)
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -9e-39) {
		tmp = d1 * (d2 - d3);
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -9e-39:
		tmp = d1 * (d2 - d3)
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -9e-39)
		tmp = Float64(d1 * Float64(d2 - d3));
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -9e-39)
		tmp = d1 * (d2 - d3);
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -9e-39], N[(d1 * N[(d2 - d3), $MachinePrecision]), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -9.0000000000000002e-39

   1. Initial program 84.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 - \left(d1 \cdot d3 + {d1}^{2}\right)} \]
   3. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \left(d1 \cdot d2 - d1 \cdot d3\right) - \color{blue}{{d1}^{2}} \]

      2. distribute-lft-out-- N/A

         \[\leadsto d1 \cdot \left(d2 - d3\right) - {\color{blue}{d1}}^{2} \]

      3. *-commutative N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - {\color{blue}{d1}}^{2} \]

      4. pow2 N/A

         \[\leadsto \left(d2 - d3\right) \cdot d1 - d1 \cdot \color{blue}{d1} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(\left(d2 - d3\right) - d1\right)} \]

      7. lower--.f64 N/A

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - \color{blue}{d1}\right) \]

      8. lower--.f64 83.5

         \[\leadsto d1 \cdot \left(\left(d2 - d3\right) - d1\right) \]

   4. Applied rewrites 83.5%

      \[\leadsto \color{blue}{d1 \cdot \left(\left(d2 - d3\right) - d1\right)} \]

   5. Taylor expanded in d1 around 0

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 69.5

         \[\leadsto d1 \cdot \left(d2 - d3\right) \]

   7. Applied rewrites 69.5%

      \[\leadsto d1 \cdot \left(d2 - \color{blue}{d3}\right) \]

   if -9.0000000000000002e-39 < d2

   1. Initial program 88.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      11. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      12. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

      14. pow2 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

      15. associate--l+ N/A

          \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d1 \cdot d3\right)\right)} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{d3 \cdot d1}\right)\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d3\right)\right) \cdot d1} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot d3\right)} \cdot d1 + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   3. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right)} \]

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot \left(d4 - d1\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 - d1\right)}\right) \]

      4. lift--.f64 74.9

         \[\leadsto d1 \cdot \left(d2 + \left(d4 - \color{blue}{d1}\right)\right) \]

   6. Applied rewrites 74.9%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d1\right)\right)} \]

   7. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   8. Step-by-step derivation

      1. lift--.f64 59.5

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   9. Applied rewrites 59.5%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -2.2 \cdot 10^{+56}:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d1 \cdot \left(d4 - d1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -2.2e+56) (* d2 d1) (* d1 (- d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2.2e+56) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-2.2d+56)) then
        tmp = d2 * d1
    else
        tmp = d1 * (d4 - d1)
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -2.2e+56) {
		tmp = d2 * d1;
	} else {
		tmp = d1 * (d4 - d1);
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -2.2e+56:
		tmp = d2 * d1
	else:
		tmp = d1 * (d4 - d1)
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -2.2e+56)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d1 * Float64(d4 - d1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -2.2e+56)
		tmp = d2 * d1;
	else
		tmp = d1 * (d4 - d1);
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -2.2e+56], N[(d2 * d1), $MachinePrecision], N[(d1 * N[(d4 - d1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -2.20000000000000016e56

   1. Initial program 82.3%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 63.1

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   4. Applied rewrites 63.1%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -2.20000000000000016e56 < d2

   1. Initial program 88.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - \color{blue}{d1 \cdot d1} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - d1 \cdot d3\right) + \color{blue}{d4 \cdot d1}\right) - d1 \cdot d1 \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right)} - d1 \cdot d1 \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{d1 \cdot d2} - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(d1 \cdot d2 - \color{blue}{d1 \cdot d3}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      7. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 - d1 \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\color{blue}{\left(d1 \cdot d2 + \left(\mathsf{neg}\left(d1\right)\right) \cdot d3\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      9. mul-1-neg N/A

         \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{\left(-1 \cdot d1\right)} \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(d1 \cdot d2 + \color{blue}{-1 \cdot \left(d1 \cdot d3\right)}\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

      11. +-commutative N/A

          \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right)} + d4 \cdot d1\right) - d1 \cdot d1 \]

      12. *-commutative N/A

          \[\leadsto \left(\left(-1 \cdot \left(d1 \cdot d3\right) + d1 \cdot d2\right) + \color{blue}{d1 \cdot d4}\right) - d1 \cdot d1 \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right)} - d1 \cdot d1 \]

      14. pow2 N/A

          \[\leadsto \left(-1 \cdot \left(d1 \cdot d3\right) + \left(d1 \cdot d2 + d1 \cdot d4\right)\right) - \color{blue}{{d1}^{2}} \]

      15. associate--l+ N/A

          \[\leadsto \color{blue}{-1 \cdot \left(d1 \cdot d3\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d1 \cdot d3\right)\right)} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{d3 \cdot d1}\right)\right) + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(d3\right)\right) \cdot d1} + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

      19. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(-1 \cdot d3\right)} \cdot d1 + \left(\left(d1 \cdot d2 + d1 \cdot d4\right) - {d1}^{2}\right) \]

   3. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-d3, d1, \mathsf{fma}\left(d2, d1, d1 \cdot \left(d4 - d1\right)\right)\right)} \]

   4. Taylor expanded in d3 around 0

      \[\leadsto \color{blue}{d1 \cdot d2 + d1 \cdot \left(d4 - d1\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto d1 \cdot \color{blue}{\left(d2 + \left(d4 - d1\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto d1 \cdot \left(d2 + \color{blue}{\left(d4 - d1\right)}\right) \]

      4. lift--.f64 74.7

         \[\leadsto d1 \cdot \left(d2 + \left(d4 - \color{blue}{d1}\right)\right) \]

   6. Applied rewrites 74.7%

      \[\leadsto \color{blue}{d1 \cdot \left(d2 + \left(d4 - d1\right)\right)} \]

   7. Taylor expanded in d2 around 0

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
   8. Step-by-step derivation

      1. lift--.f64 59.4

         \[\leadsto d1 \cdot \left(d4 - d1\right) \]

   9. Applied rewrites 59.4%

      \[\leadsto d1 \cdot \left(d4 - \color{blue}{d1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -135:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{elif}\;d2 \leq -1.3 \cdot 10^{-269}:\\ \;\;\;\;\left(-d1\right) \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -135.0) (* d2 d1) (if (<= d2 -1.3e-269) (* (- d1) d1) (* d4 d1))))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -135.0) {
		tmp = d2 * d1;
	} else if (d2 <= -1.3e-269) {
		tmp = -d1 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-135.0d0)) then
        tmp = d2 * d1
    else if (d2 <= (-1.3d-269)) then
        tmp = -d1 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -135.0) {
		tmp = d2 * d1;
	} else if (d2 <= -1.3e-269) {
		tmp = -d1 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -135.0:
		tmp = d2 * d1
	elif d2 <= -1.3e-269:
		tmp = -d1 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -135.0)
		tmp = Float64(d2 * d1);
	elseif (d2 <= -1.3e-269)
		tmp = Float64(Float64(-d1) * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -135.0)
		tmp = d2 * d1;
	elseif (d2 <= -1.3e-269)
		tmp = -d1 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -135.0], N[(d2 * d1), $MachinePrecision], If[LessEqual[d2, -1.3e-269], N[((-d1) * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d2 < -135

   1. Initial program 83.9%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 54.9

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   4. Applied rewrites 54.9%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -135 < d2 < -1.3e-269

   1. Initial program 91.1%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d1 around inf

      \[\leadsto \color{blue}{-1 \cdot {d1}^{2}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({d1}^{2}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(d1 \cdot d1\right) \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot \color{blue}{d1} \]

      4. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot d1 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot d1\right) \cdot \color{blue}{d1} \]

      6. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(d1\right)\right) \cdot d1 \]

      7. lower-neg.f64 39.9

         \[\leadsto \left(-d1\right) \cdot d1 \]

   4. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\left(-d1\right) \cdot d1} \]

   if -1.3e-269 < d2

   1. Initial program 87.8%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d4 \cdot \color{blue}{d1} \]

      2. lift-*.f64 31.5

         \[\leadsto d4 \cdot \color{blue}{d1} \]

   4. Applied rewrites 31.5%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d2 \leq -470000000:\\ \;\;\;\;d2 \cdot d1\\ \mathbf{else}:\\ \;\;\;\;d4 \cdot d1\\ \end{array} \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4)
 :precision binary64
 (if (<= d2 -470000000.0) (* d2 d1) (* d4 d1)))

C

double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -470000000.0) {
		tmp = d2 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    real(8) :: tmp
    if (d2 <= (-470000000.0d0)) then
        tmp = d2 * d1
    else
        tmp = d4 * d1
    end if
    code = tmp
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	double tmp;
	if (d2 <= -470000000.0) {
		tmp = d2 * d1;
	} else {
		tmp = d4 * d1;
	}
	return tmp;
}

Python

def code(d1, d2, d3, d4):
	tmp = 0
	if d2 <= -470000000.0:
		tmp = d2 * d1
	else:
		tmp = d4 * d1
	return tmp

Julia

function code(d1, d2, d3, d4)
	tmp = 0.0
	if (d2 <= -470000000.0)
		tmp = Float64(d2 * d1);
	else
		tmp = Float64(d4 * d1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(d1, d2, d3, d4)
	tmp = 0.0;
	if (d2 <= -470000000.0)
		tmp = d2 * d1;
	else
		tmp = d4 * d1;
	end
	tmp_2 = tmp;
end

Wolfram

code[d1_, d2_, d3_, d4_] := If[LessEqual[d2, -470000000.0], N[(d2 * d1), $MachinePrecision], N[(d4 * d1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d2 < -4.7e8

   1. Initial program 83.7%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d2 around inf

      \[\leadsto \color{blue}{d1 \cdot d2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d2 \cdot \color{blue}{d1} \]

      2. lower-*.f64 56.0

         \[\leadsto d2 \cdot \color{blue}{d1} \]

   4. Applied rewrites 56.0%

      \[\leadsto \color{blue}{d2 \cdot d1} \]

   if -4.7e8 < d2

   1. Initial program 88.8%

      \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

   2. Taylor expanded in d4 around inf

      \[\leadsto \color{blue}{d1 \cdot d4} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto d4 \cdot \color{blue}{d1} \]

      2. lift-*.f64 32.9

         \[\leadsto d4 \cdot \color{blue}{d1} \]

   4. Applied rewrites 32.9%

      \[\leadsto \color{blue}{d4 \cdot d1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 31.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ d2 \cdot d1 \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d2 d1))

C

double code(double d1, double d2, double d3, double d4) {
	return d2 * d1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d2 * d1
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d2 * d1;
}

Python

def code(d1, d2, d3, d4):
	return d2 * d1

Julia

function code(d1, d2, d3, d4)
	return Float64(d2 * d1)
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d2 * d1;
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d2 * d1), $MachinePrecision]

Derivation

1. Initial program 87.5%

   \[\left(\left(d1 \cdot d2 - d1 \cdot d3\right) + d4 \cdot d1\right) - d1 \cdot d1 \]

2. Taylor expanded in d2 around inf

   \[\leadsto \color{blue}{d1 \cdot d2} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d2 \cdot \color{blue}{d1} \]

   2. lower-*.f64 31.0

      \[\leadsto d2 \cdot \color{blue}{d1} \]

4. Applied rewrites 31.0%

   \[\leadsto \color{blue}{d2 \cdot d1} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(\left(\left(d2 - d3\right) + d4\right) - d1\right) \end{array} \]

FPCore

(FPCore (d1 d2 d3 d4) :precision binary64 (* d1 (- (+ (- d2 d3) d4) d1)))

C

double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - d1);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d1, d2, d3, d4)
use fmin_fmax_functions
    real(8), intent (in) :: d1
    real(8), intent (in) :: d2
    real(8), intent (in) :: d3
    real(8), intent (in) :: d4
    code = d1 * (((d2 - d3) + d4) - d1)
end function

Java

public static double code(double d1, double d2, double d3, double d4) {
	return d1 * (((d2 - d3) + d4) - d1);
}

Python

def code(d1, d2, d3, d4):
	return d1 * (((d2 - d3) + d4) - d1)

Julia

function code(d1, d2, d3, d4)
	return Float64(d1 * Float64(Float64(Float64(d2 - d3) + d4) - d1))
end

MATLAB

function tmp = code(d1, d2, d3, d4)
	tmp = d1 * (((d2 - d3) + d4) - d1);
end

Wolfram

code[d1_, d2_, d3_, d4_] := N[(d1 * N[(N[(N[(d2 - d3), $MachinePrecision] + d4), $MachinePrecision] - d1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025134 
(FPCore (d1 d2 d3 d4)
  :name "FastMath dist4"
  :precision binary64

  :alt
  (! :herbie-platform c (* d1 (- (+ (- d2 d3) d4) d1)))

  (- (+ (- (* d1 d2) (* d1 d3)) (* d4 d1)) (* d1 d1)))