Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.6% → 97.8%

Time: 7.1s

Alternatives: 7

Speedup: 1.1×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 - x) * z) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x) * z) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{y - x}{\frac{t}{z}} + x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((y - x) / (t / z)) + 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 = ((y - x) / (t / z)) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   6. lower-/.f64 N/A

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   7. lower-/.f64 97.7

      \[\leadsto x + \frac{y - x}{\color{blue}{\frac{t}{z}}} \]

4. Applied rewrites 97.7%

   \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

5. Final simplification 97.7%

   \[\leadsto \frac{y - x}{\frac{t}{z}} + x \]
6. Add Preprocessing

Alternative 2: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -6 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -6e-58) t_1 (if (<= t 1.45e-15) (/ (* z (- y x)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -6e-58) {
		tmp = t_1;
	} else if (t <= 1.45e-15) {
		tmp = (z * (y - x)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -6e-58)
		tmp = t_1;
	elseif (t <= 1.45e-15)
		tmp = Float64(Float64(z * Float64(y - x)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -6e-58], t$95$1, If[LessEqual[t, 1.45e-15], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.00000000000000015e-58 or 1.45000000000000009e-15 < t

   1. Initial program 85.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t}}, z, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -6.00000000000000015e-58 < t < 1.45000000000000009e-15

   1. Initial program 98.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 89.4

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

   5. Applied rewrites 89.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -6 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -6e-58) t_1 (if (<= t 1.45e-15) (* (/ z t) (- y x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -6e-58) {
		tmp = t_1;
	} else if (t <= 1.45e-15) {
		tmp = (z / t) * (y - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -6e-58)
		tmp = t_1;
	elseif (t <= 1.45e-15)
		tmp = Float64(Float64(z / t) * Float64(y - x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -6e-58], t$95$1, If[LessEqual[t, 1.45e-15], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.00000000000000015e-58 or 1.45000000000000009e-15 < t

   1. Initial program 85.6%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t}}, z, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

   if -6.00000000000000015e-58 < t < 1.45000000000000009e-15

   1. Initial program 98.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 88.5%

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

Alternative 4: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-238}:\\ \;\;\;\;\left(-x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -8.5e-186) t_1 (if (<= t 1.9e-238) (* (- x) (/ z t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -8.5e-186) {
		tmp = t_1;
	} else if (t <= 1.9e-238) {
		tmp = -x * (z / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -8.5e-186)
		tmp = t_1;
	elseif (t <= 1.9e-238)
		tmp = Float64(Float64(-x) * Float64(z / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -8.5e-186], t$95$1, If[LessEqual[t, 1.9e-238], N[((-x) * N[(z / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.4999999999999994e-186 or 1.8999999999999998e-238 < t

   1. Initial program 89.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t}}, z, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 77.5

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

   7. Applied rewrites 77.5%

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

   if -8.4999999999999994e-186 < t < 1.8999999999999998e-238

   1. Initial program 96.8%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 93.6

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

   5. Applied rewrites 93.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot z}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 68.8%

       \[\leadsto \frac{z}{t} \cdot \left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-238}:\\ \;\;\;\;\left(-x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -5e-109) t_1 (if (<= t 1.65e-96) (* (/ z t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -5e-109) {
		tmp = t_1;
	} else if (t <= 1.65e-96) {
		tmp = (z / t) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -5e-109)
		tmp = t_1;
	elseif (t <= 1.65e-96)
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -5e-109], t$95$1, If[LessEqual[t, 1.65e-96], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000002e-109 or 1.64999999999999995e-96 < t

   1. Initial program 88.1%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{t}}, z, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 81.2

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

   7. Applied rewrites 81.2%

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

   if -5.0000000000000002e-109 < t < 1.64999999999999995e-96

   1. Initial program 97.5%

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
   4. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 47.0

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 59.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 97.5

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

4. Applied rewrites 97.5%

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

Alternative 7: 39.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{t} \cdot y \end{array} \]

FPCore

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

C

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

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 = (z / t) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

   \[x + \frac{\left(y - x\right) \cdot z}{t} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{y \cdot z}{t}} \]
4. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 35.5

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

5. Applied rewrites 35.5%

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

7. Applied rewrites 39.3%

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

8. Final simplification 39.3%

   \[\leadsto \frac{z}{t} \cdot y \]
9. Add Preprocessing

Developer Target 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	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 (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

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