Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.6% → 97.9%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

• 24.9% 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: 97.9% accurate, 0.1× 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 93.6%

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

   1. +-commutative 93.6%

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

   2. *-commutative 93.6%

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

   3. associate-*l/ 98.4%

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

   4. fma-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq \infty\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* z (- y x)) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 INFINITY)))
     (+ x (* z (/ (- y x) t)))
     t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= ((double) INFINITY))) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + ((z * (y - x)) / t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= math.inf):
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(z * Float64(y - x)) / t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= Inf))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + ((z * (y - x)) / t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= Inf)))
		tmp = x + (z * ((y - x) / t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0 or +inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 78.8%

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

      1. associate-*l/ 99.9%

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

   3. Applied egg-rr 99.9%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < +inf.0

   1. Initial program 96.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y - x\right)}{t} \leq -\infty \lor \neg \left(x + \frac{z \cdot \left(y - x\right)}{t} \leq \infty\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-123} \lor \neg \left(z \leq 1.5 \cdot 10^{-231}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7e-123) (not (<= z 1.5e-231)))
   (+ x (* z (/ (- y x) t)))
   (+ x (/ (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7e-123) || !(z <= 1.5e-231)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((z * 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 <= (-1.7d-123)) .or. (.not. (z <= 1.5d-231))) then
        tmp = x + (z * ((y - x) / t))
    else
        tmp = x + ((z * 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 <= -1.7e-123) || !(z <= 1.5e-231)) {
		tmp = x + (z * ((y - x) / t));
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7e-123) or not (z <= 1.5e-231):
		tmp = x + (z * ((y - x) / t))
	else:
		tmp = x + ((z * y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7e-123) || !(z <= 1.5e-231))
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.7e-123) || ~((z <= 1.5e-231)))
		tmp = x + (z * ((y - x) / t));
	else
		tmp = x + ((z * y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7e-123], N[Not[LessEqual[z, 1.5e-231]], $MachinePrecision]], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e-123 or 1.5000000000000001e-231 < z

   1. Initial program 92.7%

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

      1. associate-*l/ 96.8%

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

   3. Applied egg-rr 96.8%

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

   if -1.7e-123 < z < 1.5000000000000001e-231

   1. Initial program 98.0%

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

   2. Taylor expanded in y around inf 90.5%

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

      1. *-commutative 90.5%

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

   4. Simplified 90.5%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-123} \lor \neg \left(z \leq 1.5 \cdot 10^{-231}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-167} \lor \neg \left(y \leq 5.6 \cdot 10^{-127}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.45e-167) (not (<= y 5.6e-127)))
   (+ x (* (/ z t) y))
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.45e-167) || !(y <= 5.6e-127)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x * (1.0 - (z / 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 <= (-2.45d-167)) .or. (.not. (y <= 5.6d-127))) then
        tmp = x + ((z / t) * y)
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.45e-167) || !(y <= 5.6e-127)) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.45e-167) or not (y <= 5.6e-127):
		tmp = x + ((z / t) * y)
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.45e-167) || !(y <= 5.6e-127))
		tmp = Float64(x + Float64(Float64(z / t) * y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.45e-167) || ~((y <= 5.6e-127)))
		tmp = x + ((z / t) * y);
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.45e-167], N[Not[LessEqual[y, 5.6e-127]], $MachinePrecision]], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.45000000000000002e-167 or 5.59999999999999999e-127 < y

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf 85.4%

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

      1. associate-*r/ 89.0%

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

   4. Simplified 89.0%

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

   if -2.45000000000000002e-167 < y < 5.59999999999999999e-127

   1. Initial program 93.5%

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

   2. Taylor expanded in x around inf 93.5%

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

      1. mul-1-neg 93.5%

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

      2. unsub-neg 93.5%

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

   4. Simplified 93.5%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-167} \lor \neg \left(y \leq 5.6 \cdot 10^{-127}\right):\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 5: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-167}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.25e-167)
   (+ x (* (/ z t) y))
   (if (<= y 6e-127) (* x (- 1.0 (/ z t))) (+ x (/ y (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-167) {
		tmp = x + ((z / t) * y);
	} else if (y <= 6e-127) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (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 (y <= (-2.25d-167)) then
        tmp = x + ((z / t) * y)
    else if (y <= 6d-127) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = x + (y / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-167) {
		tmp = x + ((z / t) * y);
	} else if (y <= 6e-127) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = x + (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.25e-167:
		tmp = x + ((z / t) * y)
	elif y <= 6e-127:
		tmp = x * (1.0 - (z / t))
	else:
		tmp = x + (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.25e-167)
		tmp = Float64(x + Float64(Float64(z / t) * y));
	elseif (y <= 6e-127)
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(x + Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.25e-167)
		tmp = x + ((z / t) * y);
	elseif (y <= 6e-127)
		tmp = x * (1.0 - (z / t));
	else
		tmp = x + (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.25e-167], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-127], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2500000000000001e-167

   1. Initial program 92.1%

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

   2. Taylor expanded in y around inf 81.6%

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

      1. associate-*r/ 84.5%

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

   4. Simplified 84.5%

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

   if -2.2500000000000001e-167 < y < 6.00000000000000017e-127

   1. Initial program 93.5%

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

   2. Taylor expanded in x around inf 93.5%

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

      1. mul-1-neg 93.5%

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

      2. unsub-neg 93.5%

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

   4. Simplified 93.5%

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

   if 6.00000000000000017e-127 < y

   1. Initial program 95.5%

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

   2. Taylor expanded in y around inf 89.7%

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

      1. associate-/l* 94.1%

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

   4. Simplified 94.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-167}:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 6: 50.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2050000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2050000000000.0) x (if (<= t 1.3e-53) (* (/ z t) (- x)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2050000000000.0) {
		tmp = x;
	} else if (t <= 1.3e-53) {
		tmp = (z / t) * -x;
	} else {
		tmp = 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 (t <= (-2050000000000.0d0)) then
        tmp = x
    else if (t <= 1.3d-53) then
        tmp = (z / t) * -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2050000000000.0) {
		tmp = x;
	} else if (t <= 1.3e-53) {
		tmp = (z / t) * -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2050000000000.0:
		tmp = x
	elif t <= 1.3e-53:
		tmp = (z / t) * -x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2050000000000.0)
		tmp = x;
	elseif (t <= 1.3e-53)
		tmp = Float64(Float64(z / t) * Float64(-x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2050000000000.0)
		tmp = x;
	elseif (t <= 1.3e-53)
		tmp = (z / t) * -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2050000000000.0], x, If[LessEqual[t, 1.3e-53], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05e12 or 1.29999999999999998e-53 < t

   1. Initial program 88.6%

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

   2. Taylor expanded in z around 0 62.8%

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

   if -2.05e12 < t < 1.29999999999999998e-53

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. mul-1-neg 55.4%

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

      2. unsub-neg 55.4%

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

   4. Simplified 55.4%

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

   5. Taylor expanded in z around inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. distribute-frac-neg 44.4%

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

   7. Simplified 44.4%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2050000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 50.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2450000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{z \cdot x}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2450000000000.0) x (if (<= t 1.8e-57) (/ (* z x) (- t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2450000000000.0) {
		tmp = x;
	} else if (t <= 1.8e-57) {
		tmp = (z * x) / -t;
	} else {
		tmp = 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 (t <= (-2450000000000.0d0)) then
        tmp = x
    else if (t <= 1.8d-57) then
        tmp = (z * x) / -t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2450000000000.0) {
		tmp = x;
	} else if (t <= 1.8e-57) {
		tmp = (z * x) / -t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2450000000000.0:
		tmp = x
	elif t <= 1.8e-57:
		tmp = (z * x) / -t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2450000000000.0)
		tmp = x;
	elseif (t <= 1.8e-57)
		tmp = Float64(Float64(z * x) / Float64(-t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2450000000000.0)
		tmp = x;
	elseif (t <= 1.8e-57)
		tmp = (z * x) / -t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2450000000000.0], x, If[LessEqual[t, 1.8e-57], N[(N[(z * x), $MachinePrecision] / (-t)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.45e12 or 1.8000000000000001e-57 < t

   1. Initial program 88.6%

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

   2. Taylor expanded in z around 0 62.8%

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

   if -2.45e12 < t < 1.8000000000000001e-57

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 55.4%

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

      1. mul-1-neg 55.4%

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

      2. unsub-neg 55.4%

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

   4. Simplified 55.4%

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

   5. Taylor expanded in z around inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. distribute-frac-neg 44.4%

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

   7. Simplified 44.4%

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

      1. frac-2neg 44.4%

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

      2. remove-double-neg 44.4%

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

      3. associate-*r/ 45.0%

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

      4. *-commutative 45.0%

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

   9. Applied egg-rr 45.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2450000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{z \cdot x}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. associate-/l* 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 9: 38.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 7.2e+214) x (* (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 7.2e+214) {
		tmp = x;
	} else {
		tmp = (z / 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 <= 7.2d+214) then
        tmp = x
    else
        tmp = (z / 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 <= 7.2e+214) {
		tmp = x;
	} else {
		tmp = (z / t) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 7.2e+214:
		tmp = x
	else:
		tmp = (z / t) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 7.2e+214)
		tmp = x;
	else
		tmp = Float64(Float64(z / t) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 7.2e+214)
		tmp = x;
	else
		tmp = (z / t) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 7.2e+214], x, N[(N[(z / t), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.2000000000000002e214

   1. Initial program 93.7%

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

   2. Taylor expanded in z around 0 39.8%

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

   if 7.2000000000000002e214 < z

   1. Initial program 92.7%

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

   2. Taylor expanded in x around inf 40.2%

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

      1. mul-1-neg 40.2%

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

      2. unsub-neg 40.2%

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

   4. Simplified 40.2%

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

   5. Taylor expanded in z around inf 38.5%

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

      1. mul-1-neg 38.5%

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

      2. distribute-frac-neg 38.5%

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

   7. Simplified 38.5%

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

      1. add-sqr-sqrt 36.8%

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

      2. sqrt-unprod 68.5%

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

      3. distribute-frac-neg 68.5%

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

      4. distribute-frac-neg 68.5%

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

      5. sqr-neg 68.5%

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

      6. sqrt-unprod 31.1%

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

      7. add-sqr-sqrt 38.8%

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

      8. associate-*r/ 31.5%

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

      9. associate-/l* 38.8%

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

   9. Applied egg-rr 38.8%

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

   10. Taylor expanded in x around 0 31.5%

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

       1. associate-*r/ 38.8%

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

       2. *-commutative 38.8%

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

   12. Simplified 38.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot x\\ \end{array} \]

Alternative 10: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 - \frac{z}{t}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- 1.0 (/ z t))))

C

double code(double x, double y, double z, double t) {
	return x * (1.0 - (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 * (1.0d0 - (z / t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * (1.0 - (z / t));
}

Python

def code(x, y, z, t):
	return x * (1.0 - (z / t))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(1.0 - Float64(z / t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * (1.0 - (z / t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in x around inf 64.0%

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

   1. mul-1-neg 64.0%

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

   2. unsub-neg 64.0%

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

4. Simplified 64.0%

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

5. Final simplification 64.0%

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

Alternative 11: 38.0% 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 93.6%

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

2. Taylor expanded in z around 0 38.0%

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

3. Final simplification 38.0%

   \[\leadsto x \]

Developer target: 97.6% accurate, 0.7× 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 2023322 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (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)))))

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