Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 10

Speedup: 1.0×

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Python

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Python

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y - z, t - x, x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (- y z) (- t x) x))

C

double code(double x, double y, double z, double t) {
	return fma((y - z), (t - x), x);
}

Julia

function code(x, y, z, t)
	return fma(Float64(y - z), Float64(t - x), x)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \left(t - x\right) + x} \]

   2. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, t - x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, t - x, x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y - z, t - x, x\right) \]

Alternative 2: 58.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ t_2 := x - z \cdot t\\ \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-234}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0.00025:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))) (t_2 (- x (* z t))))
   (if (<= z -1.05)
     t_1
     (if (<= z 5e-234)
       t_2
       (if (<= z 5.4e-94)
         (* y t)
         (if (<= z 1.12e-21) t_2 (if (<= z 0.00025) (* y t) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double t_2 = x - (z * t);
	double tmp;
	if (z <= -1.05) {
		tmp = t_1;
	} else if (z <= 5e-234) {
		tmp = t_2;
	} else if (z <= 5.4e-94) {
		tmp = y * t;
	} else if (z <= 1.12e-21) {
		tmp = t_2;
	} else if (z <= 0.00025) {
		tmp = y * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = z * (x - t)
    t_2 = x - (z * t)
    if (z <= (-1.05d0)) then
        tmp = t_1
    else if (z <= 5d-234) then
        tmp = t_2
    else if (z <= 5.4d-94) then
        tmp = y * t
    else if (z <= 1.12d-21) then
        tmp = t_2
    else if (z <= 0.00025d0) then
        tmp = y * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double t_2 = x - (z * t);
	double tmp;
	if (z <= -1.05) {
		tmp = t_1;
	} else if (z <= 5e-234) {
		tmp = t_2;
	} else if (z <= 5.4e-94) {
		tmp = y * t;
	} else if (z <= 1.12e-21) {
		tmp = t_2;
	} else if (z <= 0.00025) {
		tmp = y * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x - t)
	t_2 = x - (z * t)
	tmp = 0
	if z <= -1.05:
		tmp = t_1
	elif z <= 5e-234:
		tmp = t_2
	elif z <= 5.4e-94:
		tmp = y * t
	elif z <= 1.12e-21:
		tmp = t_2
	elif z <= 0.00025:
		tmp = y * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	t_2 = Float64(x - Float64(z * t))
	tmp = 0.0
	if (z <= -1.05)
		tmp = t_1;
	elseif (z <= 5e-234)
		tmp = t_2;
	elseif (z <= 5.4e-94)
		tmp = Float64(y * t);
	elseif (z <= 1.12e-21)
		tmp = t_2;
	elseif (z <= 0.00025)
		tmp = Float64(y * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x - t);
	t_2 = x - (z * t);
	tmp = 0.0;
	if (z <= -1.05)
		tmp = t_1;
	elseif (z <= 5e-234)
		tmp = t_2;
	elseif (z <= 5.4e-94)
		tmp = y * t;
	elseif (z <= 1.12e-21)
		tmp = t_2;
	elseif (z <= 0.00025)
		tmp = y * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05], t$95$1, If[LessEqual[z, 5e-234], t$95$2, If[LessEqual[z, 5.4e-94], N[(y * t), $MachinePrecision], If[LessEqual[z, 1.12e-21], t$95$2, If[LessEqual[z, 0.00025], N[(y * t), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.05000000000000004 or 2.5000000000000001e-4 < z

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 82.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 82.9%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 82.9%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 82.9%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 82.9%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 82.9%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 82.9%

         \[\leadsto x - z \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      2. distribute-lft-in 78.7%

         \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   6. Applied egg-rr 78.7%

      \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 82.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t + -1 \cdot x\right) \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 82.9%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot \left(t + -1 \cdot x\right)\right)} \]

      2. mul-1-neg 82.9%

         \[\leadsto -1 \cdot \left(z \cdot \left(t + \color{blue}{\left(-x\right)}\right)\right) \]

      3. sub-neg 82.9%

         \[\leadsto -1 \cdot \left(z \cdot \color{blue}{\left(t - x\right)}\right) \]

      4. associate-*r* 82.9%

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(t - x\right)} \]

      5. neg-mul-1 82.9%

         \[\leadsto \color{blue}{\left(-z\right)} \cdot \left(t - x\right) \]

      6. sub-neg 82.9%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      7. +-commutative 82.9%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(-x\right) + t\right)} \]

      8. distribute-lft-in 78.7%

         \[\leadsto \color{blue}{\left(-z\right) \cdot \left(-x\right) + \left(-z\right) \cdot t} \]

      9. distribute-rgt-neg-in 78.7%

         \[\leadsto \color{blue}{\left(-\left(-z\right) \cdot x\right)} + \left(-z\right) \cdot t \]

      10. distribute-lft-neg-in 78.7%

          \[\leadsto \color{blue}{\left(-\left(-z\right)\right) \cdot x} + \left(-z\right) \cdot t \]

      11. remove-double-neg 78.7%

          \[\leadsto \color{blue}{z} \cdot x + \left(-z\right) \cdot t \]

      12. cancel-sign-sub-inv 78.7%

          \[\leadsto \color{blue}{z \cdot x - z \cdot t} \]

      13. distribute-lft-out-- 82.9%

          \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   9. Simplified 82.9%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   if -1.05000000000000004 < z < 4.99999999999999979e-234 or 5.4000000000000002e-94 < z < 1.11999999999999998e-21

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 43.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 43.0%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 43.0%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 43.0%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 43.0%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 43.0%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]

   5. Taylor expanded in t around inf 42.4%

      \[\leadsto x - \color{blue}{t \cdot z} \]
   6. Step-by-step derivation

      1. *-commutative 42.4%

         \[\leadsto x - \color{blue}{z \cdot t} \]

   7. Simplified 42.4%

      \[\leadsto x - \color{blue}{z \cdot t} \]

   if 4.99999999999999979e-234 < z < 5.4000000000000002e-94 or 1.11999999999999998e-21 < z < 2.5000000000000001e-4

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 79.3%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in y around inf 61.8%

      \[\leadsto \color{blue}{y \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-234}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-94}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-21}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{elif}\;z \leq 0.00025:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \end{array} \]

Alternative 3: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-55}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+15}:\\ \;\;\;\;x + \left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -1.1e+15)
     t_1
     (if (<= z -3.3e-55)
       (- x (* y x))
       (if (<= z 1.16e+15) (+ x (* (- y z) t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -1.1e+15) {
		tmp = t_1;
	} else if (z <= -3.3e-55) {
		tmp = x - (y * x);
	} else if (z <= 1.16e+15) {
		tmp = x + ((y - z) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x - t)
    if (z <= (-1.1d+15)) then
        tmp = t_1
    else if (z <= (-3.3d-55)) then
        tmp = x - (y * x)
    else if (z <= 1.16d+15) then
        tmp = x + ((y - z) * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -1.1e+15) {
		tmp = t_1;
	} else if (z <= -3.3e-55) {
		tmp = x - (y * x);
	} else if (z <= 1.16e+15) {
		tmp = x + ((y - z) * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x - t)
	tmp = 0
	if z <= -1.1e+15:
		tmp = t_1
	elif z <= -3.3e-55:
		tmp = x - (y * x)
	elif z <= 1.16e+15:
		tmp = x + ((y - z) * t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -1.1e+15)
		tmp = t_1;
	elseif (z <= -3.3e-55)
		tmp = Float64(x - Float64(y * x));
	elseif (z <= 1.16e+15)
		tmp = Float64(x + Float64(Float64(y - z) * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x - t);
	tmp = 0.0;
	if (z <= -1.1e+15)
		tmp = t_1;
	elseif (z <= -3.3e-55)
		tmp = x - (y * x);
	elseif (z <= 1.16e+15)
		tmp = x + ((y - z) * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+15], t$95$1, If[LessEqual[z, -3.3e-55], N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.16e+15], N[(x + N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e15 or 1.16e15 < z

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 86.2%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 86.2%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 86.2%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 86.2%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 86.2%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 86.2%

         \[\leadsto x - z \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      2. distribute-lft-in 81.7%

         \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   6. Applied egg-rr 81.7%

      \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t + -1 \cdot x\right) \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot \left(t + -1 \cdot x\right)\right)} \]

      2. mul-1-neg 86.2%

         \[\leadsto -1 \cdot \left(z \cdot \left(t + \color{blue}{\left(-x\right)}\right)\right) \]

      3. sub-neg 86.2%

         \[\leadsto -1 \cdot \left(z \cdot \color{blue}{\left(t - x\right)}\right) \]

      4. associate-*r* 86.2%

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(t - x\right)} \]

      5. neg-mul-1 86.2%

         \[\leadsto \color{blue}{\left(-z\right)} \cdot \left(t - x\right) \]

      6. sub-neg 86.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      7. +-commutative 86.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(-x\right) + t\right)} \]

      8. distribute-lft-in 81.7%

         \[\leadsto \color{blue}{\left(-z\right) \cdot \left(-x\right) + \left(-z\right) \cdot t} \]

      9. distribute-rgt-neg-in 81.7%

         \[\leadsto \color{blue}{\left(-\left(-z\right) \cdot x\right)} + \left(-z\right) \cdot t \]

      10. distribute-lft-neg-in 81.7%

          \[\leadsto \color{blue}{\left(-\left(-z\right)\right) \cdot x} + \left(-z\right) \cdot t \]

      11. remove-double-neg 81.7%

          \[\leadsto \color{blue}{z} \cdot x + \left(-z\right) \cdot t \]

      12. cancel-sign-sub-inv 81.7%

          \[\leadsto \color{blue}{z \cdot x - z \cdot t} \]

      13. distribute-lft-out-- 86.2%

          \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   9. Simplified 86.2%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   if -1.1e15 < z < -3.2999999999999999e-55

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in z around 0 96.8%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right) + x} \]

   3. Taylor expanded in t around 0 76.7%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} + x \]
   4. Step-by-step derivation

      1. associate-*r* 76.7%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot x} + x \]

      2. *-commutative 76.7%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot y\right)} + x \]

      3. mul-1-neg 76.7%

         \[\leadsto x \cdot \color{blue}{\left(-y\right)} + x \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{x \cdot \left(-y\right)} + x \]

   if -3.2999999999999999e-55 < z < 1.16e15

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 73.4%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-55}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+15}:\\ \;\;\;\;x + \left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \end{array} \]

Alternative 4: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-221}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-92}:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -2e+16)
     t_1
     (if (<= z 6.8e-221) (- x (* y x)) (if (<= z 3e-92) (* y t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -2e+16) {
		tmp = t_1;
	} else if (z <= 6.8e-221) {
		tmp = x - (y * x);
	} else if (z <= 3e-92) {
		tmp = y * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x - t)
    if (z <= (-2d+16)) then
        tmp = t_1
    else if (z <= 6.8d-221) then
        tmp = x - (y * x)
    else if (z <= 3d-92) then
        tmp = y * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -2e+16) {
		tmp = t_1;
	} else if (z <= 6.8e-221) {
		tmp = x - (y * x);
	} else if (z <= 3e-92) {
		tmp = y * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x - t)
	tmp = 0
	if z <= -2e+16:
		tmp = t_1
	elif z <= 6.8e-221:
		tmp = x - (y * x)
	elif z <= 3e-92:
		tmp = y * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -2e+16)
		tmp = t_1;
	elseif (z <= 6.8e-221)
		tmp = Float64(x - Float64(y * x));
	elseif (z <= 3e-92)
		tmp = Float64(y * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x - t);
	tmp = 0.0;
	if (z <= -2e+16)
		tmp = t_1;
	elseif (z <= 6.8e-221)
		tmp = x - (y * x);
	elseif (z <= 3e-92)
		tmp = y * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+16], t$95$1, If[LessEqual[z, 6.8e-221], N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3e-92], N[(y * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e16 or 3.00000000000000013e-92 < z

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 81.4%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 81.4%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 81.4%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 81.4%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 81.4%

         \[\leadsto x - z \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   6. Applied egg-rr 77.5%

      \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 79.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t + -1 \cdot x\right) \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot \left(t + -1 \cdot x\right)\right)} \]

      2. mul-1-neg 79.2%

         \[\leadsto -1 \cdot \left(z \cdot \left(t + \color{blue}{\left(-x\right)}\right)\right) \]

      3. sub-neg 79.2%

         \[\leadsto -1 \cdot \left(z \cdot \color{blue}{\left(t - x\right)}\right) \]

      4. associate-*r* 79.2%

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(t - x\right)} \]

      5. neg-mul-1 79.2%

         \[\leadsto \color{blue}{\left(-z\right)} \cdot \left(t - x\right) \]

      6. sub-neg 79.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      7. +-commutative 79.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(-x\right) + t\right)} \]

      8. distribute-lft-in 75.4%

         \[\leadsto \color{blue}{\left(-z\right) \cdot \left(-x\right) + \left(-z\right) \cdot t} \]

      9. distribute-rgt-neg-in 75.4%

         \[\leadsto \color{blue}{\left(-\left(-z\right) \cdot x\right)} + \left(-z\right) \cdot t \]

      10. distribute-lft-neg-in 75.4%

          \[\leadsto \color{blue}{\left(-\left(-z\right)\right) \cdot x} + \left(-z\right) \cdot t \]

      11. remove-double-neg 75.4%

          \[\leadsto \color{blue}{z} \cdot x + \left(-z\right) \cdot t \]

      12. cancel-sign-sub-inv 75.4%

          \[\leadsto \color{blue}{z \cdot x - z \cdot t} \]

      13. distribute-lft-out-- 79.2%

          \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   9. Simplified 79.2%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   if -2e16 < z < 6.8000000000000003e-221

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in z around 0 95.6%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right) + x} \]

   3. Taylor expanded in t around 0 69.9%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} + x \]
   4. Step-by-step derivation

      1. associate-*r* 69.9%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot x} + x \]

      2. *-commutative 69.9%

         \[\leadsto \color{blue}{x \cdot \left(-1 \cdot y\right)} + x \]

      3. mul-1-neg 69.9%

         \[\leadsto x \cdot \color{blue}{\left(-y\right)} + x \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{x \cdot \left(-y\right)} + x \]

   if 6.8000000000000003e-221 < z < 3.00000000000000013e-92

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 82.0%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in y around inf 63.3%

      \[\leadsto \color{blue}{y \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-221}:\\ \;\;\;\;x - y \cdot x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-92}:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \end{array} \]

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+14} \lor \neg \left(z \leq 6.6 \cdot 10^{+18}\right):\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.6e+14) (not (<= z 6.6e+18)))
   (* z (- x t))
   (+ x (* y (- t x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+14) || !(z <= 6.6e+18)) {
		tmp = z * (x - t);
	} else {
		tmp = x + (y * (t - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.6d+14)) .or. (.not. (z <= 6.6d+18))) then
        tmp = z * (x - t)
    else
        tmp = x + (y * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+14) || !(z <= 6.6e+18)) {
		tmp = z * (x - t);
	} else {
		tmp = x + (y * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.6e+14) or not (z <= 6.6e+18):
		tmp = z * (x - t)
	else:
		tmp = x + (y * (t - x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.6e+14) || !(z <= 6.6e+18))
		tmp = Float64(z * Float64(x - t));
	else
		tmp = Float64(x + Float64(y * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.6e+14) || ~((z <= 6.6e+18)))
		tmp = z * (x - t);
	else
		tmp = x + (y * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.6e+14], N[Not[LessEqual[z, 6.6e+18]], $MachinePrecision]], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e14 or 6.6e18 < z

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 86.2%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 86.2%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 86.2%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 86.2%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 86.2%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 86.2%

         \[\leadsto x - z \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      2. distribute-lft-in 81.7%

         \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   6. Applied egg-rr 81.7%

      \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 86.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t + -1 \cdot x\right) \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 86.2%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot \left(t + -1 \cdot x\right)\right)} \]

      2. mul-1-neg 86.2%

         \[\leadsto -1 \cdot \left(z \cdot \left(t + \color{blue}{\left(-x\right)}\right)\right) \]

      3. sub-neg 86.2%

         \[\leadsto -1 \cdot \left(z \cdot \color{blue}{\left(t - x\right)}\right) \]

      4. associate-*r* 86.2%

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(t - x\right)} \]

      5. neg-mul-1 86.2%

         \[\leadsto \color{blue}{\left(-z\right)} \cdot \left(t - x\right) \]

      6. sub-neg 86.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      7. +-commutative 86.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(-x\right) + t\right)} \]

      8. distribute-lft-in 81.7%

         \[\leadsto \color{blue}{\left(-z\right) \cdot \left(-x\right) + \left(-z\right) \cdot t} \]

      9. distribute-rgt-neg-in 81.7%

         \[\leadsto \color{blue}{\left(-\left(-z\right) \cdot x\right)} + \left(-z\right) \cdot t \]

      10. distribute-lft-neg-in 81.7%

          \[\leadsto \color{blue}{\left(-\left(-z\right)\right) \cdot x} + \left(-z\right) \cdot t \]

      11. remove-double-neg 81.7%

          \[\leadsto \color{blue}{z} \cdot x + \left(-z\right) \cdot t \]

      12. cancel-sign-sub-inv 81.7%

          \[\leadsto \color{blue}{z \cdot x - z \cdot t} \]

      13. distribute-lft-out-- 86.2%

          \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   9. Simplified 86.2%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   if -2.6e14 < z < 6.6e18

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in z around 0 89.7%

      \[\leadsto \color{blue}{y \cdot \left(t - x\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+14} \lor \neg \left(z \leq 6.6 \cdot 10^{+18}\right):\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(t - x\right)\\ \end{array} \]

Alternative 6: 56.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+14} \lor \neg \left(z \leq 2.9 \cdot 10^{-92}\right):\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.7e+14) (not (<= z 2.9e-92))) (* z (- x t)) (* y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.7e+14) || !(z <= 2.9e-92)) {
		tmp = z * (x - t);
	} else {
		tmp = y * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.7d+14)) .or. (.not. (z <= 2.9d-92))) then
        tmp = z * (x - t)
    else
        tmp = y * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.7e+14) || !(z <= 2.9e-92)) {
		tmp = z * (x - t);
	} else {
		tmp = y * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.7e+14) or not (z <= 2.9e-92):
		tmp = z * (x - t)
	else:
		tmp = y * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.7e+14) || !(z <= 2.9e-92))
		tmp = Float64(z * Float64(x - t));
	else
		tmp = Float64(y * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.7e+14) || ~((z <= 2.9e-92)))
		tmp = z * (x - t);
	else
		tmp = y * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.7e+14], N[Not[LessEqual[z, 2.9e-92]], $MachinePrecision]], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], N[(y * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7e14 or 2.89999999999999985e-92 < z

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in y around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t - x\right) \cdot z\right) + x} \]
   3. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{x + -1 \cdot \left(\left(t - x\right) \cdot z\right)} \]

      2. mul-1-neg 81.4%

         \[\leadsto x + \color{blue}{\left(-\left(t - x\right) \cdot z\right)} \]

      3. unsub-neg 81.4%

         \[\leadsto \color{blue}{x - \left(t - x\right) \cdot z} \]

      4. *-commutative 81.4%

         \[\leadsto x - \color{blue}{z \cdot \left(t - x\right)} \]

   4. Simplified 81.4%

      \[\leadsto \color{blue}{x - z \cdot \left(t - x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 81.4%

         \[\leadsto x - z \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      2. distribute-lft-in 77.5%

         \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   6. Applied egg-rr 77.5%

      \[\leadsto x - \color{blue}{\left(z \cdot t + z \cdot \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 79.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(t + -1 \cdot x\right) \cdot z\right)} \]
   8. Step-by-step derivation

      1. *-commutative 79.2%

         \[\leadsto -1 \cdot \color{blue}{\left(z \cdot \left(t + -1 \cdot x\right)\right)} \]

      2. mul-1-neg 79.2%

         \[\leadsto -1 \cdot \left(z \cdot \left(t + \color{blue}{\left(-x\right)}\right)\right) \]

      3. sub-neg 79.2%

         \[\leadsto -1 \cdot \left(z \cdot \color{blue}{\left(t - x\right)}\right) \]

      4. associate-*r* 79.2%

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(t - x\right)} \]

      5. neg-mul-1 79.2%

         \[\leadsto \color{blue}{\left(-z\right)} \cdot \left(t - x\right) \]

      6. sub-neg 79.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(t + \left(-x\right)\right)} \]

      7. +-commutative 79.2%

         \[\leadsto \left(-z\right) \cdot \color{blue}{\left(\left(-x\right) + t\right)} \]

      8. distribute-lft-in 75.4%

         \[\leadsto \color{blue}{\left(-z\right) \cdot \left(-x\right) + \left(-z\right) \cdot t} \]

      9. distribute-rgt-neg-in 75.4%

         \[\leadsto \color{blue}{\left(-\left(-z\right) \cdot x\right)} + \left(-z\right) \cdot t \]

      10. distribute-lft-neg-in 75.4%

          \[\leadsto \color{blue}{\left(-\left(-z\right)\right) \cdot x} + \left(-z\right) \cdot t \]

      11. remove-double-neg 75.4%

          \[\leadsto \color{blue}{z} \cdot x + \left(-z\right) \cdot t \]

      12. cancel-sign-sub-inv 75.4%

          \[\leadsto \color{blue}{z \cdot x - z \cdot t} \]

      13. distribute-lft-out-- 79.2%

          \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   9. Simplified 79.2%

      \[\leadsto \color{blue}{z \cdot \left(x - t\right)} \]

   if -2.7e14 < z < 2.89999999999999985e-92

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 71.0%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in y around inf 40.1%

      \[\leadsto \color{blue}{y \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+14} \lor \neg \left(z \leq 2.9 \cdot 10^{-92}\right):\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \left(t - x\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - z) * (t - x))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

Python

def code(x, y, z, t):
	return x + ((y - z) * (t - x))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - z) * (t - x));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]

2. Final simplification 100.0%

   \[\leadsto x + \left(y - z\right) \cdot \left(t - x\right) \]

Alternative 8: 37.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+16} \lor \neg \left(z \leq 2.4 \cdot 10^{-92}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e+16) (not (<= z 2.4e-92))) (* z (- t)) (* y t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+16) || !(z <= 2.4e-92)) {
		tmp = z * -t;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-5.5d+16)) .or. (.not. (z <= 2.4d-92))) then
        tmp = z * -t
    else
        tmp = y * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e+16) || !(z <= 2.4e-92)) {
		tmp = z * -t;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e+16) or not (z <= 2.4e-92):
		tmp = z * -t
	else:
		tmp = y * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e+16) || !(z <= 2.4e-92))
		tmp = Float64(z * Float64(-t));
	else
		tmp = Float64(y * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.5e+16) || ~((z <= 2.4e-92)))
		tmp = z * -t;
	else
		tmp = y * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e+16], N[Not[LessEqual[z, 2.4e-92]], $MachinePrecision]], N[(z * (-t)), $MachinePrecision], N[(y * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e16 or 2.4000000000000001e-92 < z

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 55.4%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in z around inf 43.2%

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 43.2%

         \[\leadsto \color{blue}{-t \cdot z} \]

      2. distribute-rgt-neg-out 43.2%

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

   5. Simplified 43.2%

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

   if -5.5e16 < z < 2.4000000000000001e-92

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 71.0%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in y around inf 40.1%

      \[\leadsto \color{blue}{y \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+16} \lor \neg \left(z \leq 2.4 \cdot 10^{-92}\right):\\ \;\;\;\;z \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \]

Alternative 9: 37.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-17}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-17) (* y t) (if (<= y 4.8e-97) x (* y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-17) {
		tmp = y * t;
	} else if (y <= 4.8e-97) {
		tmp = x;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9d-17)) then
        tmp = y * t
    else if (y <= 4.8d-97) then
        tmp = x
    else
        tmp = y * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-17) {
		tmp = y * t;
	} else if (y <= 4.8e-97) {
		tmp = x;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-17:
		tmp = y * t
	elif y <= 4.8e-97:
		tmp = x
	else:
		tmp = y * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-17)
		tmp = Float64(y * t);
	elseif (y <= 4.8e-97)
		tmp = x;
	else
		tmp = Float64(y * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-17)
		tmp = y * t;
	elseif (y <= 4.8e-97)
		tmp = x;
	else
		tmp = y * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-17], N[(y * t), $MachinePrecision], If[LessEqual[y, 4.8e-97], x, N[(y * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.99999999999999957e-17 or 4.8e-97 < y

   1. Initial program 100.0%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 52.6%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in y around inf 34.7%

      \[\leadsto \color{blue}{y \cdot t} \]

   if -8.99999999999999957e-17 < y < 4.8e-97

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]

   2. Taylor expanded in t around inf 75.8%

      \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

   3. Taylor expanded in x around inf 29.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-17}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \]

Alternative 10: 17.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

double code(double x, double y, double z, double t) {
	return x;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

public static double code(double x, double y, double z, double t) {
	return x;
}

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]

2. Taylor expanded in t around inf 61.6%

   \[\leadsto x + \color{blue}{t \cdot \left(y - z\right)} \]

3. Taylor expanded in x around inf 14.3%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 14.3%

   \[\leadsto x \]

Developer target: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (+ (* t (- y z)) (* (- x) (- y z)))))

C

double code(double x, double y, double z, double t) {
	return x + ((t * (y - z)) + (-x * (y - z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((t * (y - z)) + (-x * (y - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((t * (y - z)) + (-x * (y - z)));
}

Python

def code(x, y, z, t):
	return x + ((t * (y - z)) + (-x * (y - z)))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(t * Float64(y - z)) + Float64(Float64(-x) * Float64(y - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((t * (y - z)) + (-x * (y - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] + N[((-x) * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

  :herbie-target
  (+ x (+ (* t (- y z)) (* (- x) (- y z))))

  (+ x (* (- y z) (- t x))))